We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonline...We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.展开更多
The article explores the issue of designing a new design of a loading cylinder with a casing filled with vulcanized rubber for pneumomechanical spinning machines. The theoretical calculation of the deformed state of a...The article explores the issue of designing a new design of a loading cylinder with a casing filled with vulcanized rubber for pneumomechanical spinning machines. The theoretical calculation of the deformed state of a cylindrical shell filled with vulcanized rubber is given. Deflections and stresses in the rubber layer are determined, which we use approximately for the Ritz methods. The theory of the radial and axial moving rubber layer was analyzed. The specific energy of deformation of a cylindrical layer of a compound cylinder is determined. The statics of the case and the loading cylinder of spinning machines are thoroughly studied.展开更多
For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitia...For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. A priori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones.展开更多
The Ritz vectors obtained by Arnoldi's method may not be good approxima- tions and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the e...The Ritz vectors obtained by Arnoldi's method may not be good approxima- tions and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m + 1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modi- fied m-step Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m + 1)-step one. Based on this strategy, we present a modified m-step restarted Arnoldi algorithm. Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard (m+ 1)-step restarted ones.展开更多
The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz ...The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz vectors, a modified strategy is proposed such that new approximate eigenvectors are certain combinations of the Ritz vectors and the waSted (m+1) th block basis vector and their corresponding residual norms are minimized in a certain sense. They can be cheaply computed by solving a few small 'dimensional minimization problems. The resulting modified m-step block Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, a modified m-step iterative block Arnoldi algorithm is presented. Numerical experiments are reported to show that the modified m-step algorithm is often considerably more efficient than the standard (m+1)-step iterative one.展开更多
This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and tw...This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.展开更多
Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical ...Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary conditions.We establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation functions.In addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples.Technically,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.展开更多
Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple ei...Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (λ, x) of a large matrix A. Given a subspace ω that contains an approximation to x, these two methods compute approximations (μ,x) and μ,x) to (λ,x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector x is unique as the deviation of x from ω approaches zero if A is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, x), we derive lower and upper bounds for the sine of the angle between the Ritz vector x and the refined eigenvector approximation x, and we prove that x≠x unless x = x. Third, we establish relationships between the residual norm ||AX -μx|| of the conventional methods and the residual norm ||Ax -μx|| of the refined methods, and we show that the latter is always smaller than the former if (μ, x) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.展开更多
Natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure are studied. The multiple layered cylindrical shell configuration is formed by three layers of isotropic mat...Natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure are studied. The multiple layered cylindrical shell configuration is formed by three layers of isotropic material where the inner and outer layers are stainless steel and the middle layer is aluminum. The multiple layered shell equations with lateral pressure are established based on Love＇s shell theory. The governing equations of motion with lateral pressure are employed by using energy functional and applying the Ritz method. The boundary conditions represented by end conditions of the multiple layered cylindrical shell are simply supported-clamped（SS-C）, free-clamped（F-C） and simply supported-free（SS-F）. The influence of different lateral pressures, different thickness to radius ratios, different length to radius ratios and effect of the asymmetric boundary conditions on natural frequency characteristics are studied. It is shown that the lateral pressure has effect on the natural frequency of multiple layered cylindrical shell and causes the natural frequency to increase. The natural frequency of the developed multilayered cylindrical shell is validated by comparing with those in the literature. The proposed research provides an effective approach for vibration analysis shell structures subjected to lateral pressure with an energy method.展开更多
A new theory developed from extended high-order sandwich panel theory(EHSAPT)is set up to assess the static response of sandwich panels by considering the geometrical and material nonlinearities simultaneously.The geo...A new theory developed from extended high-order sandwich panel theory(EHSAPT)is set up to assess the static response of sandwich panels by considering the geometrical and material nonlinearities simultaneously.The geometrical nonlinearity is considered by adopting the Green-Lagrange-type strain for the face sheets and core.The material nonlinearity is included as a piecewise function matched to the experimental stress-strain curve using a polynomial fitting technique.A Ritz technique is applied to solve the governing equations.The results show that the stress stiffening feature is well captured in the geometric nonlinear analysis.The effect of the geometric nonlinearity in the face sheets on the displacement response is more significant when the stiffness ratio of the face sheets to the core is large.The geometric nonlinearity decreases the shear stress and increases the normal stress in the sandwich core.By comparison with open literature and finite element simulations,the present nonlinear EHSAPT is shown to be sufficiently precise for estimating the nonlinear static response of sandwich beams by considering the geometric and material nonlinearities simultaneously.展开更多
Deep learning has achieved great success in solving partial differential equations(PDEs),where the loss is often defined as an integral.The accuracy and efficiency of these algorithms depend greatly on the quadrature ...Deep learning has achieved great success in solving partial differential equations(PDEs),where the loss is often defined as an integral.The accuracy and efficiency of these algorithms depend greatly on the quadrature method.We propose to apply quasi-Monte Carlo(QMC)methods to the Deep Ritz Method(DRM)for solving the Neumann problems for the Poisson equation and the static Schr¨odinger equation.For error estimation,we decompose the error of using the deep learning algorithm to solve PDEs into the generalization error,the approximation error and the training error.We establish the upper bounds and prove that QMC-based DRM achieves an asymptotically smaller error bound than DRM.Numerical experiments show that the proposed method converges faster in all cases and the variances of the gradient estimators of randomized QMC-based DRM are much smaller than those of DRM,which illustrates the superiority of QMC in deep learning over MC.展开更多
In this paper, the influence of an exponential volume fraction law on the vibration frequencies of thin functionally graded cylindrical shells is studied. Material properties in the shell thickness direction are grade...In this paper, the influence of an exponential volume fraction law on the vibration frequencies of thin functionally graded cylindrical shells is studied. Material properties in the shell thickness direction are graded in accordance with the exponential law. Expressions for the strain-displacement and curvature-displacement relationships are taken from Love＇s thin shell theory. The Rayleigh-Ritz approach is used to derive the shell eigenfrequency equation. Axial modal dependence is assumed in the characteristic beam functions. Natural frequencies of the shells are observed to be dependent on the constituent volume fractions. The results are compared with those available in the literature for the validity of the present methodology.展开更多
A nonconforming finite element method for the nonlinear parabolic equations is studied inthis paper.The convergence analysis is presented and the optimal error estimate in L^2(‖·‖_h)norm isobtained through Ritz...A nonconforming finite element method for the nonlinear parabolic equations is studied inthis paper.The convergence analysis is presented and the optimal error estimate in L^2(‖·‖_h)norm isobtained through Ritz projection technique,where ‖·‖_h is a norm over the finite element space.展开更多
The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials(FGMs)are presented.The material properties are supposed to be changed uniformly from the ceramic phase to the m...The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials(FGMs)are presented.The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness.To estimate the associated effective material properties,various homogenization schemes including the Reuss model,the Voigt model,the Mori-Tanaka model,and the Hashin-Shtrikman bound model are used.The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates.The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem.It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index.Also,by increasing the skew angle,the critical shear buckling load of an FGM skew nanoplate enhances.This pattern becomes a bit less significant for a higher value of the material property gradient index.Furthermore,among various homogenization models,the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads,and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.展开更多
Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference meth...Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference method and finite element method,the enforcement of boundary conditions in deep neural networks is highly nontrivial.One general strategy is to use the penalty method.In the work,we conduct a comparison study for elliptic problems with four different boundary conditions,i.e.,Dirichlet,Neumann,Robin,and periodic boundary conditions,using two representative methods:deep Galerkin method and deep Ritz method.In the former,the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter.Therefore,it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions.However,by a number of examples,we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides,in some cases,when the boundary condition can be implemented in an exact manner,we find that such a strategy not only provides a better approximate solution but also facilitates the training process.展开更多
This paper deals with the problem of the instability regions of a free-free flexible jointed bipartite beam under the follower and transversal forces as a realistic simulation of a two-stage aerospace structure. The a...This paper deals with the problem of the instability regions of a free-free flexible jointed bipartite beam under the follower and transversal forces as a realistic simulation of a two-stage aerospace structure. The aim of this study is to analyze the effects of the characteristics of a flexible joint on the beam instability to use maximum bearable propulsion force. A parametric study is conducted to investigate the effects of the stiffness and the location of the joint on the critical follower force by the Ritz method and the Newmark method, then to research the vibrational properties of the structure. It has been shown that the nature of instability is quite unpredictable and dependent on the stiffness and the location of the joint. The increase of the follower force or the transversal force will increase the vibration of the model and consequently cause a destructive phenomenon in the control system of the aerospace structure. Furthermore, this paper introduces a new concept of the parametric approach to analyze the characteristics effects of a flexible two-stage aerospace structure joint design.展开更多
It is discovered that the Hu-Washizu principle of elasticity with minor restrictions can be transformed to a maxminmaxmin form. Based on this alternate maxmin principle, necessary and sufficient conditions are establi...It is discovered that the Hu-Washizu principle of elasticity with minor restrictions can be transformed to a maxminmaxmin form. Based on this alternate maxmin principle, necessary and sufficient conditions are established to obtain qualitatively suitable approximate solutions with the Ritz method.展开更多
The nonlinear three-dimensional Debye screening in plasmas is investigated. A new kind of analytical equation, which is in agreement with the three-dimensional Poisson equation for the nonlinear Debye potential, is ob...The nonlinear three-dimensional Debye screening in plasmas is investigated. A new kind of analytical equation, which is in agreement with the three-dimensional Poisson equation for the nonlinear Debye potential, is obtained. It is shown that the symmetry distribution of the Debye screening in plasmas can be described by the equation.展开更多
This poper presents an approximate solution for calculating eigen-frequencies of transverse vibration of rectangular plates elastically restrained against rotation along edges. The formulae are not only very simple an...This poper presents an approximate solution for calculating eigen-frequencies of transverse vibration of rectangular plates elastically restrained against rotation along edges. The formulae are not only very simple and easily programmed but also have high accuracy. Finally, some numerical results are given and compared with other results obtained.展开更多
基金supported in part by the National Key Basic Research Program of China 2015CB856000Major Program of NNSFC under Grant 91130005,DOE Grant DE-SC0009248ONR Grant N00014-13-1-0338.
文摘We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.
文摘The article explores the issue of designing a new design of a loading cylinder with a casing filled with vulcanized rubber for pneumomechanical spinning machines. The theoretical calculation of the deformed state of a cylindrical shell filled with vulcanized rubber is given. Deflections and stresses in the rubber layer are determined, which we use approximately for the Ritz methods. The theory of the radial and axial moving rubber layer was analyzed. The specific energy of deformation of a cylindrical layer of a compound cylinder is determined. The statics of the case and the loading cylinder of spinning machines are thoroughly studied.
基金Project supported by the China State Major Key Projects for Basic Researchesthe National Natural Science Foundation of China (Grant No. 19571014)the Doctoral Program (97014113), the Foundation of Excellent Young Scholors of Ministry of Education
文摘For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. A priori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones.
基金the China State Key Project for Basic Researchesthe National Natural Science Foundation of ChinaThe Research Fund for th
文摘The Ritz vectors obtained by Arnoldi's method may not be good approxima- tions and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m + 1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modi- fied m-step Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m + 1)-step one. Based on this strategy, we present a modified m-step restarted Arnoldi algorithm. Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard (m+ 1)-step restarted ones.
文摘The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz vectors, a modified strategy is proposed such that new approximate eigenvectors are certain combinations of the Ritz vectors and the waSted (m+1) th block basis vector and their corresponding residual norms are minimized in a certain sense. They can be cheaply computed by solving a few small 'dimensional minimization problems. The resulting modified m-step block Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, a modified m-step iterative block Arnoldi algorithm is presented. Numerical experiments are reported to show that the modified m-step algorithm is often considerably more efficient than the standard (m+1)-step iterative one.
文摘This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.
基金supported by the National Key Research and Development Program of China(No.2020YFA0714200)the Science and Technology Major Project of Hubei Province under Grant 2021AAA010+2 种基金the National Science Foundation of China(Nos.12125103,12071362,11871474,11871385)the Natural Science Foundation of Hubei Province(No.2019CFA007)by the research fund of KLATASDSMOE.
文摘Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary conditions.We establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation functions.In addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples.Technically,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.
文摘Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (λ, x) of a large matrix A. Given a subspace ω that contains an approximation to x, these two methods compute approximations (μ,x) and μ,x) to (λ,x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector x is unique as the deviation of x from ω approaches zero if A is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, x), we derive lower and upper bounds for the sine of the angle between the Ritz vector x and the refined eigenvector approximation x, and we prove that x≠x unless x = x. Third, we establish relationships between the residual norm ||AX -μx|| of the conventional methods and the residual norm ||Ax -μx|| of the refined methods, and we show that the latter is always smaller than the former if (μ, x) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.
文摘Natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure are studied. The multiple layered cylindrical shell configuration is formed by three layers of isotropic material where the inner and outer layers are stainless steel and the middle layer is aluminum. The multiple layered shell equations with lateral pressure are established based on Love＇s shell theory. The governing equations of motion with lateral pressure are employed by using energy functional and applying the Ritz method. The boundary conditions represented by end conditions of the multiple layered cylindrical shell are simply supported-clamped（SS-C）, free-clamped（F-C） and simply supported-free（SS-F）. The influence of different lateral pressures, different thickness to radius ratios, different length to radius ratios and effect of the asymmetric boundary conditions on natural frequency characteristics are studied. It is shown that the lateral pressure has effect on the natural frequency of multiple layered cylindrical shell and causes the natural frequency to increase. The natural frequency of the developed multilayered cylindrical shell is validated by comparing with those in the literature. The proposed research provides an effective approach for vibration analysis shell structures subjected to lateral pressure with an energy method.
基金the National Natural Science Foundation of China(Grant 11432004).
文摘A new theory developed from extended high-order sandwich panel theory(EHSAPT)is set up to assess the static response of sandwich panels by considering the geometrical and material nonlinearities simultaneously.The geometrical nonlinearity is considered by adopting the Green-Lagrange-type strain for the face sheets and core.The material nonlinearity is included as a piecewise function matched to the experimental stress-strain curve using a polynomial fitting technique.A Ritz technique is applied to solve the governing equations.The results show that the stress stiffening feature is well captured in the geometric nonlinear analysis.The effect of the geometric nonlinearity in the face sheets on the displacement response is more significant when the stiffness ratio of the face sheets to the core is large.The geometric nonlinearity decreases the shear stress and increases the normal stress in the sandwich core.By comparison with open literature and finite element simulations,the present nonlinear EHSAPT is shown to be sufficiently precise for estimating the nonlinear static response of sandwich beams by considering the geometric and material nonlinearities simultaneously.
基金supported by the National Natural Science Foundation of China(Grant No.72071119).
文摘Deep learning has achieved great success in solving partial differential equations(PDEs),where the loss is often defined as an integral.The accuracy and efficiency of these algorithms depend greatly on the quadrature method.We propose to apply quasi-Monte Carlo(QMC)methods to the Deep Ritz Method(DRM)for solving the Neumann problems for the Poisson equation and the static Schr¨odinger equation.For error estimation,we decompose the error of using the deep learning algorithm to solve PDEs into the generalization error,the approximation error and the training error.We establish the upper bounds and prove that QMC-based DRM achieves an asymptotically smaller error bound than DRM.Numerical experiments show that the proposed method converges faster in all cases and the variances of the gradient estimators of randomized QMC-based DRM are much smaller than those of DRM,which illustrates the superiority of QMC in deep learning over MC.
文摘In this paper, the influence of an exponential volume fraction law on the vibration frequencies of thin functionally graded cylindrical shells is studied. Material properties in the shell thickness direction are graded in accordance with the exponential law. Expressions for the strain-displacement and curvature-displacement relationships are taken from Love＇s thin shell theory. The Rayleigh-Ritz approach is used to derive the shell eigenfrequency equation. Axial modal dependence is assumed in the characteristic beam functions. Natural frequencies of the shells are observed to be dependent on the constituent volume fractions. The results are compared with those available in the literature for the validity of the present methodology.
基金supported by the Natural Science Foundation of China under Grant Nos.10671184 and 10971203
文摘A nonconforming finite element method for the nonlinear parabolic equations is studied inthis paper.The convergence analysis is presented and the optimal error estimate in L^2(‖·‖_h)norm isobtained through Ritz projection technique,where ‖·‖_h is a norm over the finite element space.
文摘The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials(FGMs)are presented.The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness.To estimate the associated effective material properties,various homogenization schemes including the Reuss model,the Voigt model,the Mori-Tanaka model,and the Hashin-Shtrikman bound model are used.The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates.The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem.It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index.Also,by increasing the skew angle,the critical shear buckling load of an FGM skew nanoplate enhances.This pattern becomes a bit less significant for a higher value of the material property gradient index.Furthermore,among various homogenization models,the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads,and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.
基金the grants NSFC 11971021National Key R&D Program of China(No.2018YF645B0204404)NSFC 11501399(R.Du)。
文摘Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference method and finite element method,the enforcement of boundary conditions in deep neural networks is highly nontrivial.One general strategy is to use the penalty method.In the work,we conduct a comparison study for elliptic problems with four different boundary conditions,i.e.,Dirichlet,Neumann,Robin,and periodic boundary conditions,using two representative methods:deep Galerkin method and deep Ritz method.In the former,the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter.Therefore,it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions.However,by a number of examples,we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides,in some cases,when the boundary condition can be implemented in an exact manner,we find that such a strategy not only provides a better approximate solution but also facilitates the training process.
文摘This paper deals with the problem of the instability regions of a free-free flexible jointed bipartite beam under the follower and transversal forces as a realistic simulation of a two-stage aerospace structure. The aim of this study is to analyze the effects of the characteristics of a flexible joint on the beam instability to use maximum bearable propulsion force. A parametric study is conducted to investigate the effects of the stiffness and the location of the joint on the critical follower force by the Ritz method and the Newmark method, then to research the vibrational properties of the structure. It has been shown that the nature of instability is quite unpredictable and dependent on the stiffness and the location of the joint. The increase of the follower force or the transversal force will increase the vibration of the model and consequently cause a destructive phenomenon in the control system of the aerospace structure. Furthermore, this paper introduces a new concept of the parametric approach to analyze the characteristics effects of a flexible two-stage aerospace structure joint design.
文摘It is discovered that the Hu-Washizu principle of elasticity with minor restrictions can be transformed to a maxminmaxmin form. Based on this alternate maxmin principle, necessary and sufficient conditions are established to obtain qualitatively suitable approximate solutions with the Ritz method.
文摘The nonlinear three-dimensional Debye screening in plasmas is investigated. A new kind of analytical equation, which is in agreement with the three-dimensional Poisson equation for the nonlinear Debye potential, is obtained. It is shown that the symmetry distribution of the Debye screening in plasmas can be described by the equation.
文摘This poper presents an approximate solution for calculating eigen-frequencies of transverse vibration of rectangular plates elastically restrained against rotation along edges. The formulae are not only very simple and easily programmed but also have high accuracy. Finally, some numerical results are given and compared with other results obtained.
